PRICING k th -TO-DEFAULT SWAPS UNDER DEFAULT CONTAGION:
THE MATRIX-ANALYTIC APPROACH
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
Abstract. We study a model for default contagion in intensity-based credit risk and
its consequences for pricing portfolio credit derivatives. The model is specified through
default intensities which are assumed to be constant between defaults, but which can
jump at the times of defaults. The model is translated into a Markov jump process
which represents the default status in the credit portfolio. This makes it possible to
use matrix-analytic methods to derive computationally tractable closed-form expressions
for single-name credit default swap spreads and k th -to-default swap spreads. We ”semicalibrate” the model for portfolios (of up to 15 obligors) against market CDS spreads
and compute the corresponding k th -to-default spreads. In a numerical study based on
a synthetic portfolio of 15 telecom bonds we study a number of questions: how spreads
depend on the amount of default interaction; how the values of the underlying market
CDS-prices used for calibration influence k th -th-to default spreads; how a portfolio with
inhomogeneous recovery rates compares with a portfolio which satisfies the standard assumption of identical recovery rates; and, finally, how well k th -th-to default spreads in a
nonsymmetric portfolio can be approximated by spreads in a symmetric portfolio.
1. Introduction
In this paper we study dynamic dependence modelling in intensity-based credit risk. We
focus on the concept of default contagion and its consequences for pricing k th -to-default
swaps. The paper is an extension of Chapter 6 of the licentiate thesis [29].
Default dependency has attracted much interest during the last few years. A main
reason is the growing financial market of products whose payoffs are contingent on the
default behavior of a whole credit portfolio consisting of, for example, corporate bonds or
single-name credit default swaps (CDS-s). Example of such instruments that have gained
popularity are k th -to-default swaps and (synthetic) CDO-s. These products are designed
to manage and trade the risk of default dependencies. We refer to [6], [8], [16], [18], [29],
[41], [45] or [54] for more detailed descriptions of the instruments. Models which capture
Date: July 30, 2008.
Key words and phrases. Portfolio credit risk, intensity-based models, default dependence modelling,
default contagion, CDS, k th -to-default swaps, Markov jump processes, Matrix-analytic methods.
AMS 2000 subject classification: Primary 60J75; Secondary 60J22, 65C20, 91B28.
JEL subject classification: Primary G33, G13; Secondary C02, C63, G32.
Research supported by Jan Wallanders and Tom Hedelius Foundation and by the Swedish foundation
for Strategic Research through GMMC, the Gothenburg Mathematical Modelling Centre.
The authors would like to thank Rüdiger Frey, Jochen Backhaus, David Lando, Torgny Lindvall, Olle
Nerman, and Catalin Starica for useful comments.
1
2
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
default dependencies in a realistic way is at the core of pricing, hedging and managing such
instruments.
As the name suggest, default contagion, treats the phenomenon of how defaults can
”propagate” like a disease in a financial market (see e.g. [13]). There may be many
reasons for this kind of domino effect. For a very interesting discussion of sources of
default contagion, see pp. 1765-1768 in [39].
It is, of course, important for credit portfolio managers to have a quantitative grasp
of default contagion. This paper describes a new numerical approach to handle default
interactions. The underlying idea is the same as in [5], [7], [21], [22], which is to model
default contagion via a Markov jump process that represents the joint default status in
the credit portfolio. The main difference is that [21], [22] use time-varying parameters in
their practical examples and solve the Chapman-Kolmogorov equation by using numerical
methods for ODE-systems. In [7], the authors implement results from [5] by using Monte
Carlo simulations to calibrate and price credit derivatives.
In this article, we focus on intensities which are constant between defaults, but which may
jump at the default times. This makes it possible to obtain compact and computationally
tractable closed-form expressions for many quantities of interest, including k th -to-default
spreads. For this we use the so-called matrix-analytic approach, see e.g. [1]. From a
portfolio credit risk modeling point of view, it also turns out that this method posses
useful intuitive and practical features, both analytical and computationally. We believe
that these features in many senses are at least as attractive as the copula approach which
is current a standard for practitioners. (For a critical study of the copula approach in
financial mathematics, see [46]).
The number of articles on dynamic models for portfolio credit risk has grown exponentially during the last years. The subtopic of default contagion in intensity based models is
not an exception and has been studied in for example [3], [6], [9], [10], [12], [14], [16], [25],
[26], [32] [34], [39], [40], [41], [45], [51], [52], [53], [57], [59].
The paper [3] considers a chain where states record if obligors have defaulted or not,
and implemented this model for a basket of two bonds. The intensities in the model
were calibrated to market data using linear regression. In [14] the authors model default
contagion in symmetric portfolio by using a piecewise-deterministic Markov process and
find the default distribution. The book [41], pp. 126-128, studies a Markov chain model
for two firms that undergo default contagion. Further, [59] treats default contagion using
the total hazard construction of [50], [55], as first suggested in [15]. This method allows for
general time dependent and stochastic intensities and that the intensities are functionals of
the default times. The latter seems difficult to handle in a Markov jump process framework.
Given the parameters of the model, the total hazard method gives a way to simulate default
events. The total hazard construction seems rather complicated to implement even in
simple cases such as piece-wise deterministic intensities considered in this paper.
The paper [39] assumes a so called primary-secondary structure, were obligors are divided
into two groups called primary obligors and secondary obligors. The idea is that the defaultintensities of primary obligors only depend on macroeconomic market variables while the
default intensity for secondary obligors can depend on both the macroeconomic variables
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
3
and on the default status of the primary firms, but not on the default status of the other
secondary firms. Assuming this structure, [39] derives closed formulas for defaultable
bonds, default swaps, etc, also for stochastic intensities. In the article [10] the authors
propose a method where one can value defaultable claims without having to use the so
called ”no-jump condition”. This technique is then applied to find survival distributions
for a portfolio of two obligors that undergo default contagion. In [57] the author studies
counterparty risk in CDS valuation by using a four state Markov process that includes
contagion effects. [57] considers time dependent intensities and then uses perturbation
techniques to approximately solve the Chapman-Kolmogorov equation. The framework
in [57] is similar to [12], where the author treats the same problem in a setup where the
intensities are constant.
The rest of this paper is organized as follows. In Section 2 we give a short introduction
to pricing of credit k th -to-default swaps. Section 3 contains the formal definition of default
contagion used in this paper, given in terms of default intensities. It is then used to
construct such default times as hitting times of a Markov jump process.
In Section 5 we use the results of Section 4, for numerical investigation of a number
of properties of k th -to-default spreads. Specifically, we semi-calibrate portfolios with up
to 15 obligors against market CDS spreads and then compute the corresponding k th -todefault spreads. The results are used to illustrate how k th -to-default spreads depend on the
strength of default interaction, on the underlying market CDS-prices used for calibration,
and on the amount of inhomogeneity in the portfolios.
Section 6 discusses numerical issues and some possible extensions, and the final section,
Section 7 summarizes and discusses the results.
2. Pricing k th -to-default swap spreads
In this section and in the sequel all computations are assumed to be made under a
risk-neutral martingale measure P. Typically such a P exists if we rule out arbitrage
opportunities.
Consider a k th -to-default swap with maturity T where the reference entity is a basket
of m bonds, or obligors, with default times τ1 , τ2 , . . . , τm and recovery rates φ1 , φ2 , . . . , φm .
Further, let T1 < . . . < Tk be the ordering of τ1 , τ2 , . . . , τm . For k th -to-default swaps, it is
standard to let the notional amount on each bond in the portfolio have the same value,
say, N, so we assume this is the case.
The protection buyer A pays a periodic fee Rk N∆n to the protection seller B, up to
the time of the k-th default Tk , or to the time T , whichever comes first. The payments are
made at times 0 < t1 < t2 < . . . < tn = T . Further let ∆j = tj − tj−1 denote the times
between payments (measured in fractions of a year). Furthermore, if default happens for
some Tk ∈ [tj , tj+1 ], A will also pay B the accrued default premium up to Tk . On the
other hand, if Tk < T , B pays A the loss occurred at Tk , that is N(1 − φi ) if it was obligor
i which defaulted at time Tk .
The constant Rk , often called k-th-to default spread, is expressed in bp per annum and
determined so that the expected discounted cash-flows between A and B coincide at t = 0.
4
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
This implies that Rk is given by
Pm
E
1
D(T
)(1
−
φ
)1
k
i
{T
≤T
}
{T
=τ
}
i
k
k
i=1
,
Rk = Pn
(2.1)
E
D(t
)∆
1
+
D(T
)
(T
−
t
j
j {Tk >tj }
k
k
j−1 ) 1{tj−1 <τ ≤tj }
j=1
R
T
where D(T ) = exp − 0 rs ds , and rt is the so called short term risk-free interest rate
at time t. Note that N does not enter into this expression. Thus, we will from now on
without loss of generality let N = 1 when discussing spreads on credit swaps.
In the credit derivative literature today, it is standard to assume that the default times
and the short time riskfree interest rate are mutually independent, and that the recovery
rates are deterministic. Under these assumptions Equation (2.1) can be simplified to
RT
Pm
i=1 (1 − φi ) 0 B(s)dFk,i (s)
(2.2)
Rk = P R tj
n
B(t
)∆
(1
−
F
(t
))
+
B(s)
(s
−
t
)
dF
(s)
j
j
k j
j−1
k
j=1
tj−1
where B(t) = E [D(t)] is the expected value of the discount factor, and Fk (t) = P [Tk ≤ t]
and Fk,i (t) = P [Tk ≤ t, Tk = τi ] are the distribution functions of the ordered default times,
and the probability that the k-th default is by obligor i and that it occurs before t, respectively. It may be noted that in the special case when all recovery
rates are the same, say
RT
φi = φ the denominator in (2.2) can be simplified to (1 − φ) 0 B(s)dFk (s), and hence the
Fk,i are not needed in this case.
The latter of course in particular holds if there is only one bond (or obligor) so that
m = 1. This case gives the most liquidly traded instrument, called a single-name Credit
Default Swap (CDS), which has special importance in this paper as our main calibration
tool.
3. Intensity based models reinterpreted as Markov jump processes
In this section we define the intensity-based model for default contagion which is used
throughout the paper. The model is then reinterpreted in terms of a Markov jump process.
This interpretation makes it possible to use a matrix-analytic approach to derive computationally tractable closed-form expressions for single-name CDS spreads and k-th-to default
spreads. These matrix analytic methods has largely been developed for queueing theory
and reliability applications, and in these context are often called phase-type distributions,
or multivariate phase-type distributions in the case of several components (see e.g. [2]).
With τ1 , τ2 . . . , τm default times as above, define the point process Nt,i = 1{τi ≤t} and
introduce the filtrations
m
_
Ft,i = σ (Ns,i; s ≤ t) , Ft =
Ft,i .
i=1
Let λt,i be the Ft -intensity of the point processes Nt,i . Below, we will for convenience often
omit the filtration and just write intensity or ”default intensity”. With a further extension
of language we will sometimes also write that the default times {τi } have intensities {λt,i }.
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
5
The model studied in this paper is specified by requiring that the default intensities have
the following form,
X
bi,j 1{τj ≤t} ,
t ≤ τi ,
(3.1)
λt,i = ai +
j6=i
and λt,i = 0 for t > τi . Further, ai ≥ 0 and bi,j are constants such that λt,i is non-negative.
The financial interpretation of (3.1) is that the default intensities are constant, except at
the times when defaults occur: then the default intensity for obligor i jumps by an amount
bi,j if it is obligor j which has defaulted. Thus a positive bi,j means that obligor i is put
at higher risk by the default of obligor j, while a negative bi,j means that obligor i in fact
benefits from the default of j, and finally bi,j = 0 if obligor i is unaffected by the default
of j.
The intensities in Equation (3.1) only depend on which obligors that have defaulted, and
not by the order in which the defaults have occurred. Thus it is a model for Unordered
Default Contagion. A more general case is when the intensities also are affected by the
order in which defaults have happened. The approach outlined below works equally well
for such Ordered Default Contagion. We make some further comments on this at the end
of the present section.
Equation (3.1) determines the default times through their intensities. However, the
expressions (2.1) and (2.2) for the k th -to-default spreads are in terms of their joint distributions. It is by no means obvious how to go from one to the other. Here we will use the
following observation.
Proposition 3.1. There exists a Markov jump process (Yt )t≥0 on a finite state space E
and a family of sets {∆i }m
i=1 such that the stopping times
τi = inf {t > 0 : Yt ∈ ∆i } ,
i = 1, 2, . . . , m,
(3.2)
have intensities (3.1). Hence, any distribution derived from the multivariate stochastic
vector (τ1 , τ2 , . . . , τm ) can be obtained from {Yt }t≥0 .
In this paper, Proposition 3.1 is throughout used for computing distributions. However,
we still use Equation (3.1) to describe the dependencies in a credit portfolio since it is
more compact and intuitive. Proposition 3.1 is rather obvious, and perhaps most easily
understood by examples, see below. However, we still give one possible formal construction,
since it provides notation which anyhow is needed later on.
Proof of Proposition 3.1. We construct the state space as a union,
m
[
E=
Ek,
(3.3)
k=0
where E k is set of states consisting of precisely k elements of {1, . . . m},
E k = {j = {j1 , . . . jk } :
1 ≤ ji ≤ m,
i = 1, . . . k} .
(3.4)
for k = 1, . . . m, and where E 0 = {0}. The interpretation is that on the set E 0 no obligors
have defaulted, on {j1 , . . . jk } the obligors in the set have defaulted, and on E m all obligors
have defaulted.
6
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
The Markov jump process (Yt )t≥0 on E is specified by making {1, . . . m} absorbing and
starting Y in {0}, and by specifying its intensity matrix Q. The latter specification is that
transitions are only possible from E k to E k+1 , and that for a state j = {j1 , j2 , . . . , jk } ∈
E k a transition can only occur to a state j ′ = (j, jk+1) ∈ E k+1 where jk+1 6= ji for
i = 1, 2, . . . , k. Further, the intensity for transitions from j = {j1 , j2 , . . . , jk } ∈ E k to such
a j ′ is
k
X
Qj,j′ = ajk+1 +
bjk+1 ,ji .
(3.5)
i=1
The diagonal elements of Q is determined by the requirement that the row sums of an
intensity matrix is zero.
Next, set
∆i = {j ∈ E : jn = i for some jn ∈ j}
and define the hitting times τ1 , . . . , τm by
τi = inf {t > 0 : Yt ∈ ∆i } .
(3.6)
This construction is illustrated in Figure 1 for the case m = 3. It is clear from the
construction that τ1 , . . . , τm have the intensities (3.1), see e.g. [35], Chapter 4. As a final
a2 + b2,1
{1, 2}
{1}
a
a1
+
1
a3
b
a3
1,2
+
b3
,1
+
+
b3
,2
b3
,1
{0}
a2
{1, 3}
{2}
a1
a3
+
a3
b
+
a2 + b2,3
{1, 2, 3}
1,3
a1
b3
,2
{3}
a2 + b2,1 + b2,3
+b
1,
+b
3
1,2
{2, 3}
Figure 1. Illustration of the construction for m = 3. Arrows indicate possible transitions, and the transition intensities are given on top of the
arrows.
aside, when we write down Q as a matrix it is computationally convenient to order the
states in E so that Q is upper triangular. This can be done by letting {0} be first, then
taking the states in E 1 in some arbitrary order, followed by the states in E 2 in some
arbitrary order, and so on.
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
7
Table 1. The number of states in the unordered and ordered case for different
number of obligors, m.
m
5
6
7
8
9
10
unordered
32
64
128
256
512
1024
ordered
326
1957
13700
109601
986410
9864101
So far we have considered Unordered Default Contagion. In Ordered Default Contagion,
also the order in which the defaults occur influence default intensities. In our setup, this
corresponds to changing the form (3.1) of the intensities to
X
λt,i = ai +
bi,j 1nτ <...<τ ≤to , τi ≥ t,
(3.7)
j∈P
j1
j|j |
and λt,i = 0 when t > τi . Here P contains all the ordered subsets j = j1 , . . . , j|j| of the
set {1, 2, . . . , m}. Furthermore, ai and bi,j are constants such that λt,i ≥ 0.
It is easy to see that the construction for Proposition 3.1 can be extended to the case
(3.7). The basic change which has to be made is to change E k from the set of all subsets
of size k of {1, . . . m} to the set of all ordered subsets of size k.
Changing from unordered to ordered default contagion however increases the number of
states in E violently. For unordered default contagion
|E| = 2m
while for ordered default contagion
m
.
n!
|E| =
n
n=0
m
X
Table 1 shows the number of states in the unordered respectively ordered case for different
sizes of the number m of obligors.
It is of course up to the modeler to decide if it is appropriate to use ordered or unordered
default contagion. However, from the table we see that in practice it is mainly convenient
to work with unordered default contagion. Further, if possible one should for large m try
to reduce the number of states in E further, for example by using symmetries.
4. The matrix-analytic method
We now use the matrix analytic method, see e.g. [1] to find expressions for Fk (t) =
P [Tk ≤ t] and Fk,i (t) = P [Tk ≤ t, Tk = τi ], the distribution functions of the ordered default
times, and the probability that the k-th default is by obligor i and that it occurs before t.
The first one is more or less standard, while the second one is less so. These expressions
8
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
in turn give possibilities to compute the quantities which are at the center of interest
in this paper, the k th -to-default spreads. Our development is closely related to so-called
multivariate phase type distributions, see e.g. [2].
Define the probability vector p (t) = (P [Yt = j])j∈E and let α = (1, 0, . . . , 0) ∈ R|E|
be the initial distribution of the Markov jump process and let its generator be Q. From
Markov theory we know that
p (t) = αeQt ,
and P [Yt = j] = αeQt ej ,
(4.1)
where ej ∈ R|E| is a column vector where the entry at position j is 1 and the other entries
are zero. Furthermore, eQt is the matrix exponential which has a closed form expression
in terms of the eigenvalue decomposition of Q.
Next, define vectors m(k) of length |E| by requiring that
k−1
1 if
j ∈ ∪i=0
Ei
(k)
(4.2)
mj =
0 otherwise .
Then,
P [Tk > t] = αeQt m(k) ,
(4.3)
(k)
since m sums the probabilities of states where there has been less than k defaults.
Hence, what is left to compute is P [Tk > t, Tk = τi ]. For this we use the imbedded
m
Markov chain (YTn )m
n=0 . By definition, the transition probability matrix P for (YTn )n=0 is
given by
Qj,j′
P j,j′ = P YTn = j ′ | YTn−1 = j = P
, j, j ′ ∈ E,
k6=j Qj,k
with the ordering of the states in P the same as for Q.
Further, let hi,k be vectors of length |E| and let Gi,k be |E| × |E| diagonal matrices,
defined by
1 if
j ∈ ∆i ∩ E k
i,k
hj =
0 otherwise ,
and
Gi,k
j,j
=
1 if
j ∈ ∆C
i ∩ Ek
0 otherwise .
We now establish the following result.
Proposition 4.1. With notation as above,
P [Tk > t, Tk = τi ] = αeQt
k−1
k−1
X
Y
ℓ=0
p=ℓ
Gi,p P
!
hi,k ,
(4.4)
for k = 1, . . . m.
Proof of Proposition 4.1. We will use the following fact, which is straightforward to establish, and standard in Markov chain theory:
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
9
If {Xn } is a stationary, discrete time, finite state space, Markov chain with initial distribution p and transition matrix P , and E 0 , . . . E k are subsets of the state space, then
P [X0 ∈ E 0 , . . . Xk ∈ E k ] = pG0 P G1 P . . . Gk−1 P hk ,
where the Gℓ -s are diagonal matrices with diagonal elements equal to one for state j if
j ∈ E ℓ and zero otherwise, for ℓ = 0, . . . k − 1, and hk is a column vector with a 1 in
position j if j ∈ E k .
Let ∆C
i be the complement of ∆i in E, i.e. the set of all states where i has not defaulted.
By an appropriate translation to the situation and notation above, in particular replacing
p by αeQt , and if Tℓ ≤ t < Tℓ+1 < Tk , we obtain that
C
C
P Yt ∈ ∆C
i ∩ E ℓ , YTℓ+1 ∈ ∆i ∩ E ℓ+1 , . . . YTk−1 ∈ ∆i ∩ E k−1 , YTk ∈ ∆i ∩ Ek
= αeQt Gi,ℓ P Gi,ℓ+1 . . . Gi,k−1P hi,k .
Since
P [Tk > t, Tk = τi ]
k−1
X
=
ℓ=0
this proves (4.4).
C
C
P Yt ∈ ∆C
i ∩ E ℓ , YTℓ+1 ∈ ∆i ∩ E ℓ+1 , . . . YTk−1 ∈ ∆i ∩ E k−1 , YTk ∈ ∆i ∩ Ek ,
5. Numerical studies
In this section we will use the theory developed in previous sections to study, in a realistic
numerical example, how different factors affect the size of k-th to default spreads. For this
it is convenient to reparameterize the basic description (3.1) of the default intensities to
the form
!
m
X
λt,i = ai 1 + c
θi,j 1{τj ≤t} ,
(5.1)
j=1,j6=i
which was suggested in [20]. In this parametrization, the ai are the base default intensities,
c measures the general ”interaction level” and the θi,j measure the ”relative dependence
structure”.
First, in Subsection 5.1 we introduce a portfolio consisting of 15 telecom companies
which is used as a basis for the numerical studies. We further ”semi-calibrate” our model
to this portfolio, using CDS spreads taken from Reuters.
We then study the influence of portfolio size on k th -to-default spreads, (Subsection 5.2),
of changing the interaction level (Subsection 5.3), the impact of using inhomogeneous
recovery rates (Subsection 5.4), the sensitivity to the underlying CDS spreads (Subsection
5.5), and finally compare a model with non-symmetric dependence to a corresponding
symmetric model (Subsection 5.6).
For the rest of this paper we will assume that the θi,j are given - hence the term ”semicalibrations”, cf. Subsection 5.1. It is a topic for future research to find out how to
estimate the θi,j . For example, using liquid market data on CDO’s will give us more
10
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
information which can be used for some cases. The rapidly increasing market of credit
portfolio products may also help. In Section 7 we discuss this topic in more detail.
Numerical studies always carry the risk of programming errors and numerical instability. However, fortunately we have been able to benchmark our numerical methods to an
example from [19], pp. 19-20 and [20], which as far as we know, are the only available
results on default contagion for nonsymmetric portfolios with more than three bonds.
The paper [19] studies a portfolio with five bonds and time-dependent default intensities,
and uses numerical solution of differential equations to compute spreads for a number of
cases. Our model for intensities doesn’t directly allow for time-dependence, but it was still
possible to approximate the portfolio in [19] with our model, calibrate it as discussed in
Section 6 below, and compare the spreads thus obtained with those in [20]. The results
agreed to at least four significant digits in all cases, which lends some confidence to our
numerical implementation.
5.1. A telecom portfolio. Table 2 describes the portfolio which is used in our numerical
studies. The data was obtained from Reuters at August 23, 2005. We have assumed a
fictive recovery rate structure and also a fictive relative dependence structure θi,j which is
given in Table 10 in Appendix, and we used the interaction level c = 0.5. The interest rate
was assumed to be constant and set to 3%, and the protection fees were assumed to be
paid quarterly. The maturity was 5 years. The ai -s are obtained by individual calibration
to the CDS spreads in Table 2. From Table 10 we see that the intensities can jump up
to 284% of their ”base values” ai , when c = 0.5. In case both bid and ask prices for the
CDS-s were given, we used their average. The calibration is described in more detail in
Section 6. We refer to the entire procedure - using the fictive recovery rates, the fictive
dependence structure and the calibrated base intensities - as semi-calibration.
5.2. Dependence on portfolio size. To study the dependence on portfolio size, we
considered 6 different sub-portfolios. The first portfolio consisted of the 10 first bonds
from Table 2, the second of the 11 first bonds, and so on, until the last portfolio which
contained all the 15 bonds in the table. Each subportfolio with m obligors had a dependence
structure given by upper left m × m submatrix of the matrix given in Table 10. When
we calibrated the subportfolios against the market CDS spreads, the corresponding sum
of the absolute calibration error never exceeded two hundreds of a bp. For each portfolio
the k th -to-default spreads were computed from Equation (2.2). The results are shown in
Table 3. The spreads are only shown for k ≤ 5. The remaining spreads were all less than
six hundreds of a basis point.
In the table the spreads increase as the size of the portfolio increases, as they should.
Quantitatively, the increase from a portfolio of size 10 to one of size 15 is 47% for a 1st to-default swap, 92% for a 2nd -to-default swap, 168% for a 3rd -to-default swap, and for
a 5th -to-default swap the increase is 700%. Further, for a portfolio of size 10 the price
of a 1st -to-default swap is about 4500 times higher than for a 5th -to-default swap. The
corresponding ratio for a portfolio of size 15 is about 1100.
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
11
Table 2. The Telecom companies and their 5 year CDS spreads.
bid ask
time
recovery %
Company
British Telecom
40 44 23 Aug, 09:33
32%
Deutsche Telecom 34
23 Aug, 19:18
48%
Ericsson
54 54 23 Aug, 18:27
45%
38 42 23 Aug, 17:13
34%
France Telecom
Nokia
21 23 23 Aug, 12:25
42%
Hellenic Telecom
43 23 Aug, 19:18
41%
Telefonica
34 38 23 Aug, 09:34
29%
26
23 Aug, 12:25
39%
Telenor
Telecom Italia
47
23 Aug, 19:34
51%
Telia
35 23 Aug, 12:25
41%
Port Telecom Int 34 38 23 Aug, 12:10
47%
47 23 Aug, 16:29
33%
MM02
Vodafone
24 28 23 Aug, 12:59
35%
KPN
38 42 23 Aug, 09:33
43%
Telekom Aus
35 04 Aug, 19:59
50%
Table 3. The k-th-to default swap premiums in basis points (bp). The first
column is for the 10 first obligors in Table 2, the second is for the 11
first obligors, and so on.
k
1
2
3
4
5
m = 10
357.7
55.38
7.649
0.8698
0.08026
m = 11
389.8
65.27
9.963
1.281
0.1373
m = 12
432.3
77.48
12.84
1.814
0.2167
m = 13
456.6
84.34
14.49
2.132
0.2678
m = 14
493.3
95.96
17.47
2.744
0.3701
m = 15
526.1
106.8
20.40
3.366
0.4795
5.3. Dependence on the interaction level. In this subsection we use a portfolio consisting of the 9 first obligors in Table 2 to study how spreads are affected by the interaction
parameter c which was taken to be 0.5 in the previous section. As above, we let the dependence parameters be given by the upper left part of Table 10. We first note that by
(5.1) the value of c enters into the calibration of the base intensities ai : a higher value of c
will lead to smaller ai -s, and hence c affects spreads both directly, and indirectly through
its influence on the base intensities.
The dependence of the spreads on the interaction level is illustrated in Figure 2. The
1st -to-default spread decreases with increasing interaction level, and for k larger than 2
the spreads increase. However, it looks as if the 2nd -to-default spread may have a local
maxima. To confirm that this is indeed possible, we experimented with different dependence structures. One result was Figure 3, which depicts the same graphs as Figure 2 but
12
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
for different θi,j , given by Table 9 where some of elements θi,j are much bigger than the
corresponding numbers in Table 10. In this figure, the 2nd -to-default spread has a clear
local maximum.
It might be worth noting that the graph of the 1st -to-default spread as a function of
the interaction level c, roughly had the same structure as the corresponding graph for
the intensity for T1 , see Figure 6. However, the same was not true
the 2nd -to-default
Pfor
9
spread. This can be seen from Figure 7 which shows the intensity i=1,i6=j ai (1 + cθi,j ) for
the second default T2 , when the first default was by obligor j, for j = 1, 2, . . . 9.
The case c = 0 is of special interest since it means that the defaults are independent
of one-another. In particular, Figures 2, 3 and 4 quantifies the errors made in computing
spreads as if obligors where independent in cases where there in fact is default contagion.
Further, in Figure 3 we note that for very large interaction levels, the spreads for 1 ≤
k ≤ 5 tend to converge into a narrow interval, compared with the case with very small
interactions. The intuitive explanation for this may be that once one obligor default,
several other will quickly follow. Finally, note that as the interaction level increases, the
spreads for 6 ≤ k ≤ 9 drastically increases and can thus no longer be neglected (see Figure
4), as for example in the Table 3 where c = 0.5.
k−th−to default spread as a function of interaction level c
350
1st−to default spread
2nd−to default spread
300
3rd−to default spread
k−th−to default spreads (in bp)
4th−to default spread
5th−to default spread
250
200
1st
150
100
2nd
50
0
0
3rd
4th
5th
2
4
6
8
10
interaction level, c
12
14
16
18
Figure 2. The kth -to-default spreads as a function of the interaction level c, for
a portfolio consisting of the first 9 obligors in Table 2.
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
k−th−to default spread as a function of interaction level c
350
1st−to default spread
2nd−to default spread
300
rd
k−th−to default spreads (in bp)
3 −to default spread
4th−to default spread
250
5th−to default spread
200
150
100
st
1
2nd
rd
3th
4th
5
50
0
0
2
4
6
8
10
interaction level, c
12
14
16
18
Figure 3. The different kth -to-default spreads for k ≤ 5 as a function of the
interaction level c, for a portfolio consisting of the first 9 obligors in
Table 2 with dependence structure given by Table 9.
k−th−to default spread as a function of interaction level c
30
th
6 −to default spread
th
6th
7 −to default spread
25
th
k−th−to default spreads (in bp)
8 −to default spread
th
9 −to default spread
20
7th
15
8th
10
5
th
9
0
0
2
4
6
8
10
interaction level, c
12
14
16
18
Figure 4. The different kth -to-default spreads for 6 ≤ k ≤ 9 as a function of the
interaction level c, for a portfolio consisting of the first 9 obligors in
Table 2 with dependence structure given by Table 9.
13
14
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
a as a function of c
i
0.01
a1(c)
0.009
a2(c)
a3(c)
a4(c)
0.007
a5(c)
0.006
a7(c)
a6(c)
a8(c)
i
calibrated a parameters
0.008
a9(c)
0.005
0.004
a
3
0.003
a
9
6
0.002
a
a4
2
a
0.001
a
a
a785
0
0
1
2
4
6
8
10
interaction level, c
12
14
16
18
Figure 5. The base intensities ai as functions of the interaction level c for a
portfolio consisting of the first 9 obligors in Table 2 with dependence
structure given by Table 9.
Σ9i=1ai as a function of c
0.06
Σ9i=1ai(c)
0.055
i
sum of calibrated a parameters
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0
2
4
6
8
10
interaction level, c
12
14
16
18
Figure 6. The intensity for T1 , as a function of the interaction level c, for a
portfolio consisting of the 9 first obligors in Table 2 with dependence
structure given by Table 9.
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
15
9
intensities for T2 when T1=τj, i.e. Σi=1,i ≠ jai(1+cθi,j)
1.4
j= 1
j= 2
1.2
j= 3
j= 4
j
j= 6
j= 7
1
intensities for T when T =τ
=8
2
jj =
j=7
j= 5
1
jj =
=1
6
j= 8
0.8
j=4
2
j= 9
j=3
0.6
jj =
=5
9
0.4
0.2
0
0
2
4
6
8
10
interaction level, c
12
14
16
18
Figure 7. Intensities for T2 when the first default was by obligor j, for a portfolio
consisting of the 9 first obligors in Table 2 with dependence structure
given by Table 9.
Table 4. The relative difference in percent between kth -to-default swap spreads
priced with homogeneous recovery rates and nonhomogeneous recovery
rates. The different recovery rates are displayed in Table 8 in Appendix
8
k std = 3.24 std = 4.90 std = 6.92 std = 11.10
1
0.13
0.16
0.20
0.23
2
1.27
1.35
2.05
2.74
2.81
3.25
5.13
6.71
3
4
4.27
5.17
8.45
11.10
5
5.60
6.99
11.69
15.51
5.4. Dependence on the recovery rates. In this subsection the numerical experiment
aimed at investigating to what extent k th -to-default swaps spreads for portfolios with nonhomogeneous recovery rates differ from the spreads in a corresponding portfolio where all
recovery rates are the same and equal to the average of the nonhomogeneous rates.
The experiment was performed on a portfolio consisting of the first 11 obligors in Table
2 with dependence structure given by the upper left 11 × 11 submatrix of the matrix
given in Table 10. We studied five different cases. In the first one all recovery rates were
set to 40%. In the other four cases the recovery rates were varied ”randomly” around
approximately the mean 40%, but with the different standard deviations 3.24, 4.90, 6.92
16
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
and 11.10, respectively. The results are displayed in Table 4. For the 1st - and 2nd -todefault spreads, the inhomogeneous cases differed from the homogenous one by at most
3%, and even the largest difference, for k = 5 and for the standard deviation equal to
11.10%, was only 15%. The different recoveries are displayed in Table 8 in Appendix 8
5.5. Dependence on the market spreads. In this subsection we investigated in a numerical experiments how the k th -to-default swap prices change when the market prices of
the underlying single-name CDS prices change.
Table 5. kth -to-default spreads R1 , . . . R5 as function of market CDS spreads.
The portfolio consisted of the first 5 obligors in Table 2. The CDS
spreads R(2) and R(5) for obligors 2 and 5 were varied, while the other
CDS spreads were held constant. To save space we only show 3 values
for R(2) .
R
R
(2)
(2)
R(2)
R(2)
R(2)
R1
15
105
215
R(5)
15
162
248
353
65
206
291
396
105
241
326
431
145
276
361
466
215
340
424
529
R2
15
105
215
R(5)
15
10.9
23.5
36.7
65
21.3
35.9
51.6
105
28.9
45.2
62.9
145
35.9
53.9
73.6
215
47.0
67.8
91.0
R3
15
105
215
R(5)
15
0.54
1.44
2.36
65
1.53
3.28
5.10
105
2.20
4.58
7.07
145
2.78
5.75
8.85
215
3.63
7.52
11.6
R(5)
R4
15
65
105
145
215
15 0.017 0.057 0.083 0.105 0.134
105 0.061 0.192 0.280 0.354 0.460
215 0.105 0.328 0.477 0.605 0.787
15
65
105
145
215
R5
15 0.000 0.001 0.001 0.002 0.002
105 0.001 0.005 0.008 0.010 0.012
215 0.003 0.010 0.014 0.017 0.022
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
17
The experiment used a portfolio consisting of the 5 first obligors in Table 2. We used a
dependence structure given by Table 10 and interaction level c = 0.5. The CDS spreads
for obligors 1, 3, 4 were held at their market values, while the CDS spreads for obligors 2
and 5 were increased in small steps from 15 to 215. The resulting k th -to-default spreads
increased smoothly as the CDS spreads increased, and this increase was more dramatic for
larger k-s. A few of these default spreads are shown in Table 5.
5.6. Approximation by a symmetric portfolio. A portfolio is symmetric if the obligors
are completely interchangeable. In the intensity formulation (5.1) for unordered default
contagion, this means that the parameters ai all are equal, and similarly the θi,j are the
same for each obligor and the φi are equal. To compute spreads it then is sufficient to
keep track of how many obligors have defaulted, but there is no need to know which ones
it was. Here we consider the special case of (3.1) where all obligors have the same default
P
intensities λt,i = λt specified by λt = a + m−1
k=1 bk 1{Tk ≤t} where {Tk } is the ordering of
the default times {τi }. For this symmetric case, the Markov jump process constructed in
Proposition 3.1 can be collapsed into a chain with the m + 1 states {0}, {1}, . . . {m}. The
interpretation is that the chain is in the state {k} if precisely k obligors have defaulted.
In the literature such a process is called a death process. This new state space is very
much smaller state than the one in Proposition 3.1, which means that it is possible to do
numerical computation for much larger portfolios, with hundreds or thousands of obligors,
see e.g. [31] where CDO-tranche spreads are computed on portfolios with 125 obligors in
such a symmetric model.
It is hence of interest to understand how well non-symmetric portfolios can be approximated by symmetric ones. To explore this we constructed and semi-calibrated three different portfolios, 1) the telecom portfolio from Table 2 with all recovery rates set to 40%,
2) the telecom portfolio with the recovery rates given in Table 2 (the standard deviation of
these rates are 6.88%), and 3) the telecom portfolio with the first 11 recovery rates given
by the last row in Table 8 and the remaining rates for obligors 12 to 15 set to 30, 32, 40
and 60% (the rates then have standard deviation 11.71%). In all cases the dependence
structure was given by Table 10 and the interaction level was c = 0.5.
Each of these three portfolios was then approximated by a symmetric portfolio. In the
symmetric approximating portfolio all recovery rates were set equal to the average of the
recovery rates in the original portfolio. The jump parameters where assumed to be the
same so bk = b at each default time Tk . Further, the parameters a and b were chosen so
that the model CDS spread in the symmetric portfolio coincided with the average market
CDS spread in the nonsymmetric counterpart, and so that the first-to-default spreads also
agreed.
From Table 6 we see that the relative differences are small. For k = 2 and 3 they
increase as the standard deviation of the recovery rates increase. However, for k = 4 and
5 the relative differences for the second portfolio (std = 6.88) somewhat surprisingly are
smaller than for the other two cases. For the nonhomogeneous recovery rate cases, the
relative differences are not monotone in k. Finally, for 6 ≤ k ≤ 15 the differences can
18
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
range between 9% up to 130% where the error on average increases with k. However, for
each case the sum of these spreads are smaller than one tenth of a bp.
Table 6. The difference in kth -to-default swap spreads between an approximating
symmetric portfolio and a nonsymmetric portfolio, in percent of the
spreads for the nonsymmetric portfolio; for three different cases. For
k = 1 the difference is by construction (almost) zero.
k std = 0 std = 6.88 std = 11.71
2 0.153
0.587
1.98
3 0.811
0.850
3.37
4
2.24
0.339
3.82
4.65
1.26
3.13
5
6. Calibration and numerical implementation
In this section we discuss the computational aspects in more detail. Subsection 6.1 gives
a short description how to calibrate (or semi-calibrate) the model (5.1) against market data
and formulas for the CDS and k th -to-default spreads, and Subsection 6.2 discusses how to
compute the matrix exponential. Subsections 6.3 and 6.4 consider more general models
and computation by simulation, respectively.
6.1. Calibration. As discussed above, we assume that the relative dependence structure
θi,j , the interaction level c and the recovery rate structure φ1 , . . . , φm all are exogenously
given - hence the term ”semi-calibration”. The base intensities ai are then obtained by
individual calibration to the market CDS spreads. The calibration uses a nonlinear least
squares method where the model CDS spreads R(i) are matched against the corresponding
market CDS spreads Ri,M . Thus the base intensities are computed as
m
X
2
(a1 , . . . , am ) = argmin
Ri,M − R(i) (â)
â=(â1 ,...,âm ) i=1
where we have emphasized that the R(i) are functions of the parameters a = (a1 , . . . , am ).
We then use the calibrated base intensities ai with {θi,j , φi, c} to compute the k th -to default
spreads Rk .
Reverting to the notation of (3.1), closed-form expressions for Rk and R(i) may be
obtained by inserting (4.3) and (4.4) into (2.2). For ease of reference we exhibit the
resulting formulas (detailed proofs can be found in [29] or [30]).
Proposition 6.1. Consider m obligors with default intensities (3.1) and assume that the
interest rate r is constant. Then,
Rk =
α
P
n
α (A(0) − A(T )) φ(k)
Qtj e−rtj + C(t
(∆
e
,
t
))
m(k)
j
j−1
j
j=1
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
and
R(i) =
Here
φ(k) =
m
X
i=1
and
(1 − φi )α (A(0) − A(T )) g (i)
P
.
n
Qtj e−rtj + C(t
(i)
α
(∆
e
,
t
))
g
j
j−1 j
j=1
(1 − φi ) Ri,k hi,k
and
Ri,k =
k−1
k−1
Y
X
ℓ=0
Gi,p P
p=ℓ
19
!
C(s, t) = s (A(t) − A(s)) − B(t) + B(s),
where
A(t) = eQt (Q − rI)−1 Qe−rt
B(t) = eQt tI + (Q − rI)−1 (Q − rI)−1 Qe−rt
Finally, g (i) is an |E| column vector such that
1 if
j ∈ ∆C
(i)
i
gj =
0 otherwise .
There are several possible computational shortcuts. The quantities g (i) , hi,k , Gi,k and
m(k) do not depend on the parametrization,
once.
P and hence only have to be computed
n
Qtj −rtj
The row vectors α (A(0) − A(T )) and α
e
+ C(tj−1 , tj ) are the same
j=1 ∆j e
for all CDS spreads and all k th -to-default swap spreads and hence only have to be computed once for each parametrization {ai , θi,j , φi, c}. The same holds for
φ(k) for each k.
Pn
Qtj −rtj
+ C(tj−1 , tj ) can be simplified
e
Furthermore, if ∆j is constant then α
j=1 ∆j e
in terms of eQtj . If the φi are the same for all obligors, φ(k) can be replaced by (1 − φ)m(k) .
Pk−1 i,ℓ,k
Qk−1 i,p
Next, if we rewrite the sum Ri,k as Ri,k = ℓ=0
M
where M i,ℓ,k = p=ℓ
G P then
i,ℓ−1,k
i,ℓ−1
i,ℓ,k
M
= G
PM
which is useful for computation. Also note that (Q − rI) is
invertible since it is upper diagonal with strictly negative diagonal elements. The conditioning number of (Q − rI) is often large, but still we have not encountered numerical
problems in computing the inverse.
6.2. Computation of the matrix exponential. The main challenge in Proposition 6.1
is to compute the matrix exponential eQt . We have mainly experimented with two different
numerical methods; direct series expansion of the matrix exponential and the uniformization method (which sometimes also is called the randomization method). Both were fast
and robust for our problems. However, while the series expansion method lacks lower
bounds on possible worst case scenarios (see [47]), the uniformization method provides
analytical expressions for the residual error. Furthermore, previous studies indicate that it
can handle large sparse matrices with remarkable robustness, see e.g. [56], [27] or Appendix
C.2.2 in [41]. We have therefore chosen the uniformization method. A probabilistic interpretation of the method can be found in [27] and pure matrix arguments which motivates
the method are given in [56] and Appendix C.2.2 in [41].
20
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
There are many different methods to compute the matrix exponential ([47] [48]). However, most of the other standard methods are not adapted to very large, but sparse, matrices
and don’t seem possible when the state space is larger than a few hundred (see [56],[27]).
As one example, it is tempting to try eigenvalue decomposition since the eigenvalues are
given by the diagonal of Q. However, in our examples this method failed already for m = 9
since the eigenvector matrices turned out to be ill-conditioned, which introduced large numerical errors in the inversions. Compared to e.g. Krylov-based methods or ODE methods
the uniformization method is relatively simple to implement.
The uniformization method works as follows. Let Λ = max |Qj,j | : j ∈ E and set
Pe = Q/Λ + I. Then
∞
X
n
(Λt)n
Qt
e =
Pe e−Λt
.
(6.2.1)
n!
n=0
P
e n −Λt (Λt)n . Let ε > 0 be arbitrary
Recall that p (t) = αeQt and define pe (t, N) = α N
n=0 P e
n!
PN (ε) −Λt (Λt)n
< ε. Then the L1 error kp (t) − pe (t, N(ε))k1
and pick N(ε) so that 1 − n=0 e
n!
is also less than ε, i.e., pe (t, N(ε)) approximates p (t) with an accumulated absolute error
which is less than ε. Furthermore, since all entries in pe (t, N(ε)) are positive there are no
cancelation effects and the approximation error decreases monotonically with increasing
N. Furthermore, for fixed N, the error kp (t) − pe (t, N)k1 is also decreasing in t.
n
Further useful properties of (6.2.1) are that it separates the computation of Pe from the
time dependent components and that
Z T
∞
X
n Λn (Λ+r)
(Q−rI)t
e
dt =
Pe
In
(T ),
(6.2.2)
n!
0
n=0
where
In(β) (t)
=
Z
t
sn e−βs ds
0
=
(β)
e−βt (−βt)n − n(−βt)n−1 + n(n − 1)(−βt)n−2 − . . . + (−1)n−1 (n − 1)!(−βt) .
n+1
(−β)
Since In (t) > 0, there are no cancelation effects in the approximation of the RHS in
(6.2.2). Truncating the sum in the RHS in (6.2.2) gives an approximation to the integral
in the LHS. In this case, the error control requires a little more work.
For m ≤ 13, we used a standard laptop with 1024 MB RAM. For m = 14 and m = 15
the memory requirements were too big for the laptop so the computations were done on
a Sun Solaris, 2x900 MHz UltraSPARC-III with 5GB RAM. As an example if m = 15,
c = 0.5, T = 5 and θi,j were as in Table 10 and if we put ε = 3.33 · 10−16 (the floatingpoint relative accuracy for the number 1 in Matlab is 2.22 · 10−16 ) then our calibration
against Table 2 implied that Λ = 0.2500 and that N(ε) = 19 terms were needed in the
computation of pe (T, N). Furthermore, the calibration errors were negligible: the sum
of the individual absolute calibration errors were less than two tenths of a bp. More
information on computation time is given in Table 7.
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
21
Table 7. Initialization time for creating the datastructures, calibration time, and
allocated memory for a portfolio with m obligors. All computations
were made with Matlab.
m
8
9
10
11
12
13
14
15
initialization time (sec) 0.400 1.232 4.309 15.82 58.92 221.7 1087 4204
calibration time (sec) 2.170 5.173 12.84 33.23 91.93 253.4 1320 4456
allocated memory (MB) 1.168 3.196 9.022 26.10 77.01 231.0 877.7 2718
A further point is that our matrices in general are very large, for example if m = 15 then
the generator has 215 = 32768 rows and thus contain 230 ≈ 1. billion entries. However,
at the same time it is extremely sparse and the sparseness is increasing with m. E.g., for
m = 15 there are only 0.025% nonzero entries in Q, and hence only about 280.000 elements
have to be stored.
A final point is that we are not interested in finding the matrix exponential itself, but
only the probability vector p(t). This is important, since computing eQt is very time and
memory consuming compared with computing αeQt . For example, using the uniformization
method with t = 5, m = 14 and ε = 3.331 · 10−16 , which implies that N(ε) = 19 the time
to find p(t) in Matlab, by first computing eQt with the uniformization method and then
multiplying this matrix with α, was 42.3 seconds. On the other hand, computing p(t) via
n
the vectors αPe only took 0.14 seconds, and hence was about 300 times faster.
To get the same accuracy with direct Taylor method required 32 terms in the truncated
sum. For this case, the corresponding computational times where 108 seconds and 0.25
seconds. The quick method was about 450 times faster than the slow one. Further, the
uniformization method was about 2.5 and 1.8 times faster compared to the corresponding
slow and quick Taylor methods. The reason was that the Taylor method required 32 terms
in the sums, compared to the 19 needed for the uniformization method.
6.3. Extensions. It is natural to believe that the default intensity of a obligor in addition
to dependence on defaults of other obligors also depends on exogenous macroeconomic
and market factors. It is possible generalize Proposition 3.1 in Section 3 to the case
when the parameters ait and bi,j
t in the formula (3.1) for the intensity depend on some
background random process. The idea is to first condition on the whole realization of the
background process and then treat (Yt )t≥0 as an inhomogeneous Markov jump process with
time-dependent generator. It is usually impossible to find tractable closed-form expressions
for distributions derived from (τ1 , . . . , τm ) for this case. Instead, to use our method one
has to rely on a fine discretisation of time.
We also note that Proposition 3.1 does not seem easily applicable if the parameters in the
intensities depend of the default times. For example, so called self-exciting point processes
(or Hawkes processes), see e.g. [28] with intensities given by
X
bi,j e−(t−τj ) 1{τj ≤t}
(6.3.1)
λt,i = ai +
j6=i
22
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
are therefore not suited to our setup. Applications of Hawkes processes in credit risk are
discussed in e.g. [24], [23] and [17].
6.4. Simulation. An alternative to numerical computation is to use simulation to produce
realizations of the random vector (τ1 , τ2 . . . , τm ). If the intensities are given by (3.1) this
is straightforward, see e.g. [35].
The so called total hazard construction ([50], [55]) can be used in more general circumstances where the intensities may be functionals of the default times as well as of time, see
e.g. (6.3.1). The paper, [15], we believe, was the first to point out that the total hazard
construction can be used in credit risk. In Chapter 5 of [59], the author used the totalhazard construction to study four cases of default contagion. Three of these cases can be
handled without invoking the total-hazard construction, for example using our approach,
cf also [50]. The fourth case, which considers stochastic parameters, can actually also be
treated without the total hazard construction. The case with a stochastic processes X t in
the parameters as in [59], is handled by doing a straightforward extension of the results in
[50], [55].
To simulate one needs to know the values of the parameters. It is far from trivial how to
estimate the parameters in general. Although one can repeat the simulations with different
parameters until the model is calibrated, this is often very time consuming.
7. Discussion and conclusions
In this paper we considered the intensity based default contagion model (3.1), where the
default intensity of one firm is allowed to change when other firms default. The model was
reinterpreted in terms of a Markov jump process, and this reinterpretation made it possible to derive closed form expressions for k th -to-default spreads. With the computational
resources available to us these expressions were tractable for general portfolios with up to
15 obligors. These are much larger than the general examples treated by other authors.
We used a synthetic telecom portfolio with 15 companies taken from Reuters at August
23, 2005 in a numerical study of how default contagion influences k th -to-default spreads.
For this we performed a ”semi-calibration” of the portfolios where interaction parameters,
interest rates and recovery rates were assumed obtained from prior knowledge but where
the baseline default intensities were calibrated to market CDS spreads. The questions we
tried to illustrate were:
How did the size of the portfolio influence k th -to-default spreads? In our example, the
1th -to-default spread increased by about 50% when the portfolio size increased from 10 to
15 and by about 700% for a 5th -to-default spread. For a portfolio of size 10 the prize of
a 1st -to-default spread was about 4500 times higher than for a 5th -to-default spread and
for a portfolio of size 15 the spread was about 1100 times higher. Qualitatively this is
completely as expected. However it would seem rather impossible to guess the sizes of the
effects without computation.
How were k th -to-default spreads influenced by the strength of interaction in the portfolio? We considered two examples. In those the 1th -to-default spread decreased when the
interaction became stronger, and the higher order than 2nd -to-default spreads decreased.
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
23
In the first example there was some indication that 2nd -to-default spreads first increased
and then decreased, and this was clear in the second example. This last finding may be
somewhat counterintuitive. A possible intuitive explanation may be that as the interaction
increases, the 2nd -to-default will have a behavior more like the 1th -to-default spread. This
view is also supported by nothing that when the interaction level is extremely big, all swap
spreads tend to converge into a narrow interval, compared with the case with very small
interaction, see e.g. Figure 3. The results also illustrate the error made if one assumes
independence in cases where there in fact is default contagion.
How did k th -to-default spreads depend on the underlying market CDS spreads? The k th to-default spreads increased smoothly with increasing market spreads. The increase was
greater for larger k-s. This first result was complectly as expected, and the second one
perhaps slightly less obvious.
How were spreads affected by non-homogeneities in the recovery rates? The spreads were
virtually unchanged by moderate inhomogeneities in the recovery rates. This agrees with
earlier findings for single-name CDS-s, see e.g. [33].
Does approximation of a non-homogeneous portfolio with a homogenous one work well?
In our example the approximation worked quite well. The interest of this comparison is
that the computational burden for a homogenous portfolios is very much smaller than for
non-homogeneous ones.
As a general comment, qualitatively most of the results summarized above were as was
expected beforehand. However, it seems difficult to guess the sizes of the effects without
actually doing the computations.
How can we estimate the dependence structure ? There are several possible ways to
estimate, or calibrate the dependence matrix θ. One approach is to use historical time series
data on the traded CDS spreads and consider the quadratic covariance process between
the obligors i and j to get an indication of θi,j . A similar procedure can also be done
on the corresponding bonds. Another approach is to estimate equity correlation and use
this as a proxy for default correlation. To be more specific, using Proposition 3.1 it is
straightforward to find computational tractable closed-form expression for pairwise default
correlation (i.e. Corr(1{τi ≤t} , 1{τj ≤t} )) as function of time, the matrix θ, c and the baseline
intensities ai . These analytical expressions for the default correlations can be used to
extract θ from the numerical values on the corresponding equity correlations. However,
further assumptions have to be done since the equity correlations are only half as many
as the entities in the matrix θ. Using equity correlation as a proxy for default correlation
has previously been very common when modelling default dependencies, see e.g. [4], [36],
[42], [43], [44] and [49]. For example, in [42], the authors value k th -to-default swaps by
using two different copulas for a portfolio with 6 obligors. First, a one-factor Gaussian
copula is used where the six correlations are estimated from equity returns. The one-factor
Gaussian copula lacks correlations in the tails and [42] redoes the computations with a
Clayton copula where the lower tail dependence is estimated from equity returns by using
Kendall’s tau on this data. Similar techniques are also pursued in e.g. [4] and [43].
A third approach to extract θ is first to estimate default correlation in an intensity
based model from historical corporate data under the statistical probability measure. This
24
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
is done in e.g. [11], [37] and [38]. Then, if we know the relationship between the statistical
probability measure and the risk-neutral martingale measure, this can be used to extract
θ. Determining the relationship between these two measures is equivalent to finding the
connection between the intensities when changing the measures. Such a procedure is far
from trivial and a discussion of how this can be done is given in e.g. [5], [6], [45] and [58].
As discussed above the approach of this paper can be generalized to time-dependent and
stochastic intensities. Further, it also gives a possibility to price other products, such as
CDO-s. This will be presented in further papers by the first author.
To conclude, this paper gives an intuitive and tractable model for computing k th -todefault spreads. The method makes it possible to simplify numerical computation by
using analytic calculation and to sometimes make computation possible in cases which are
beyond reach of competing models.
We have been able to calibrate general portfolios with 15 obligors, and up to, say, 20
obligors may be within reach. However, due to the combinatorial explosion of the size of
the state spaces, general portfolios much larger than 20 most likely will be beyond reach for
the method. As for other methods, to make reasonable parameter choices may still be the
most difficult problem for very large general portfolios. We believe that for large problems
various simplifications, such as introducing symmetries in the portfolios or to split up into
smaller subportfolios will continue to be at the center of interest for us and for others.
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8. Appendix
In this appendix we display the recovery rate cases and the dependence structures used
in Section 5.
Table 8. Different recovery rate cases used in Table 4 and Table 6. Recoveries
expressed in %.
φ1
36
33
32
27
φ2
43
47
48
52
φ3
45
46
45
41
φ4
37
34
34
31
φ5
41
44
42
45
φ6
44
41
41
31
φ7
38
36
29
29
φ8
37
39
39
35
φ9
43
42
51
52
φ10
39
39
41
41
φ11
43
46
47
61
mean
40.55
40.64
40.82
40.45
std
3.24
4.90
6.93
11.10
DEFAULT CONTAGION IN PORTFOLIO CREDIT RISK
27
Table 9. The θ matrix describing the alternative dependence structure, rounded
to two decimal places.
i
1
2
3
4
5
6
7
8
9
j=1
0
3.84
1.84
4.18
0.35
4
4.91
6.74
5.25
j=2
3.03
0
5.62
0.59
6.62
6.41
4.22
1.78
6.11
j=3
5.13
2.96
0
0.51
3.88
2.05
6.01
2.35
4.76
j=4
2.5
3.49
3.04
0
4.32
0.8
6.29
5.28
5.54
j=5
4.69
1.41
1.92
4.39
0
0.42
0.63
1.9
2.87
j=6
6.36
4.18
2.31
3.35
4.18
0
5.81
6.69
4.17
j=7
5.69
4.27
4.91
0.65
2.98
2.63
0
5.83
5.87
j=8
6.7
1.03
6.09
5.39
3.11
4.35
6.66
0
1.74
j=9
1.32
3.44
2.87
3.25
4.28
0.5
2.2
4.26
0
(Alexander Herbertsson), Centre For Finance, Department of Economics, Göteborg
School of Business, Economics and Law, Göteborg University. P.O Box 640, SE-405 30
Göteborg, Sweden
E-mail address: Alexander.Herbertsson@economics.gu.se
(Holger Rootzén), Department of Mathematical Statistics, Chalmers University of
Technology, SE-412 96 Göteborg, Sweden
E-mail address: rootzen@math.chalmers.se
ALEXANDER HERBERTSSON AND HOLGER ROOTZÉN
28
Table 10. The θ matrix describing the dependence structure, rounded to two decimal places. This matrix is
used in all examples except in Figure 3 to Figure 7.
i
j=1
j=2
j=3
j=4
j=5
j=6
j=7
j=8
j=9
j = 10
j = 11
j = 12
j = 13
j = 14
j = 15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
0.15
0.13
2.5
0.1
0.1
0.19
1.57
0.18
1.62
1.59
0.1
0.19
0.5
0.2
5.68
0
0.16
4.79
0.13
0.13
0.19
0.17
2.46
2.45
1.52
0.11
0.69
0.15
0.33
0.11
2.91
0
0.17
0.18
0.16
0.12
2.8
0.11
1.29
0.12
0.84
0.12
0.94
0.16
0.12
2.22
0.11
0
0.18
0.2
0.13
0.15
2.76
0.12
1.28
0.12
0.41
0.16
0.16
0.13
0.14
4.61
0.14
0
0.19
0.2
2.94
0.13
1.76
1.05
1.08
0.12
0.27
0.1
0.16
0.2
5.21
0.13
0.2
0
0.17
0.17
2.5
2.47
0.87
0.14
0.19
0.2
0.4
0.11
0.16
0.18
2.11
0.16
0.14
0
3.36
0.14
1.87
0.13
0.8
0.11
0.74
0.15
0.16
0.11
0.16
1.83
0.16
0.16
0.14
0
0.9
0.18
0.71
0.18
0.64
0.12
0.19
0.13
0.2
0.11
0.14
3.87
0.11
0.1
2.04
0
0.13
0.11
1.51
0.17
0.18
0.14
0.1
0.1
0.13
0.16
1.29
0.17
0.17
0.16
2.3
0
2.15
0.14
1.32
0.12
0.14
0.12
0.15
0.13
0.17
0.13
1.87
0.1
2.6
0.13
1.72
0
0.16
0.12
0.39
0.15
0.12
0.11
0.15
0.12
0.17
3.28
0.18
0.18
2.22
2.6
0.15
0
0.17
0.13
0.57
0.13
0.14
0.19
0.17
0.2
0.17
2.42
1.19
0.16
1.44
0.13
0.2
0
0.7
0.14
0.14
0.19
0.14
0.11
0.11
0.17
2.8
0.18
1.92
0.14
0.58
0.15
0.76
0
0.17
0.15
0.14
0.16
0.19
0.19
0.12
0.15
1.38
0.15
0.18
1.4
1.14
0.11
0.16
0